How to get vertex snapping to snap to one vertex
I think what you are wanting to do is make those two segments, the handle and base, into one solid object. 

Well, the easiest way I would recommend doing this is to select the two objects in Object mode, then apply a boolean modifier set to union. This will instantly merge the two objects together for you. 

Here's a tutorial that helped me understand using boolean modifiers better if you aren't familiar with them: https://www.youtube.com/watch?v=pcUe-ab_jQs